Maximal functions associated with Fourier multipliers of Mikhlin-H"ormander type

Abstract

We show that maximal operators formed by dilations of Mikhlin-H"ormander multipliers are typically not bounded on $L^p(R^d)$. We also give rather weak conditions in terms of the decay of such multipliers under which $L^p$ boundedness of the maximal operators holds.

Final version with improved differentiability assumptions on the multipliers. To appear in Mathematische Zeitschrift